Semantics of Object Oriented Databases
Starting from fundamental structural features of OODBs such as the
distinction between values and objects and consequently between types and
classes it is argued that the semantics of OODBs can be founded on the
semantics of the underlying type system. Given a suitable logic associated
with the type system this even comprises object dynamics by a logical
characterization of operations (methods).
Type systems are considered in a very general sense allowing arbitrary
constructors to be defined. If higher-order functions come into play, type
semantics can be based on the notion of a topos. In this case the associated
logic becomes intuitionistic, but allows classical axiomatic semantics to be
generalized.

Towards a better theory of interpretations, or: maybe we need something more general than categories
One great advantage of the use of categories in logic,
apparent already in Lawvere's thesis, is that it allows for a unified
treatment of the concept of an interpretation of one theory into
another, and that of a semantics of a theory. In both case what we
end up with is a structure-preserving functor, the categorical
structure being preserved being "the logic" we are interested in.
Until quite recently the categorical structure in question was always
decribable in terms of universal properties; in other words it was a set
of properties that were obeyed by the relevant categories, as opposed
to additional defined structure.
Recently new developments in logic and computer science have made the
framework above seem too constricted. For example Linear Logic has
forced us to consider monoidal categories, which cannot be defined (in
general) by universal properties. There are also new kinds of
interpretations such that the functors seem to be more than functors,
or even monoidal functors. One very telling example is Girard's sytem
LC for classical logic. So far all attempts at giving a categorical
treatment of it have failed, and we are now sure that the reason for
this is that we need something more general than categories to
algebraicize this logic. Thus we propose a tentative new framework,
which can be tought of as a generalization of multicategories. One
interesting thing about it is that the operation of linear negation is
always there as a primitive, if only in a purely formal fashion.

Monadic Semantics of CCS-like calculi
We start from a calculus as simple as pure CCS and aim at
something like the higher-order pi-calculus. In the process
several orthogonal issues need to be addressed. As desiderata we
would like to have a parametrized semantics, which can deal with
different versions of CCS by instantiation of the parameter.

The limit-colimit coincidence in synthetic domain theory
In synthetic domain theory one starts off with an ambient
category of "sets" (typically a topos) and extracts from it a full
subcategory of "well-behaved" sets to act as predomains. In current
work, John Longley and I are investigating a notion, which we call
well-completeness, that allows a uniform definition of such a subcategory
in any realizability topos in which a single axiom holds. The examples
include the effective topos, toposes based on domain-theoretic models
of the untyped lambda-calculus, and toposes based on partial combinatory
algebras of lambda terms.
It turns out that the well-complete objects support much of standard
domain theory with one important exception: the usual statement of the
limit-colimit coincidence fails (both externally - which is hardly
surprising - and internally - which is more interesting). However, a
suitable (internal) reformulation of the limit-colimit coincidence
does hold, and can be used in the usual way to obtain solutions to
recursive domain equations as bilimits.

Covariant Types
The covariant type system is rich enough to represent
polymorphism on inductive types, such as lists and trees, and yet is
simple enough to have a set-theoretic semantics. Its chief novelty is
to replace function types by transformation types, which denote
parametric functions. Their free type variables are all in positive
positions, and so can be modelled by covariant functors. They have a
simple semantics based on data functors, which can be interpreted in
the category of sets.